Dynamics of $\lambda$-continued fractions and $\beta$-shifts

TL;DR

This paper introduces a transformation related to $eta$-shifts and $eta$-continued fractions, providing a geometric interpretation, an expansion algorithm, and analyzing the properties of the associated map.

Contribution

It establishes a conjugacy between the $eta$-shift and a new transformation $T_ ext{lambda}$ linked to $eta$-continued fractions, with detailed properties of the $eta( ext{lambda})$ map.

Findings

Defined a transformation $T_ ext{lambda}$ for $eta$-continued fractions.

Proved conjugacy between $T_ ext{lambda}$ and $eta$-shift.

Analyzed the properties of the map $eta( ext{lambda})$, including monotonicity and continuity.

Abstract

For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_\lambda$ provides an algorithm to expand any positive real number in $\lambda$-continued fraction. We prove the conjugacy between $T_\lambda$ and some $\beta$-shift, $\beta>1$. Some properties of the map $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.